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Full text of "Treatise On Analysis Vol-Ii"

7    INTEGRABLE FUNCTIONS AND SETS        123

it follows that n*(\ |/| - \u\ |) ^ e, which by (13.7.2) shows that |/| is integ-
rable. Moreover, we have -|/| ^/^ |/| and hence

A**(- I/I) = -MI/I) ^ /**(/) = X/) ^ 0*(

which establishes (13.7.4.1). Since /+ = K/ + I/I) and/" = i(|/l -/), it
follows that/+ and/" are integrable (13.7.3). Iff and # are integrable, then
f—gis defined almost everywhere and integrable, and the functions (every-
where defined) sup(/, g) and inf(/, g) are equivalent respectively to the func-
tions (defined almost everywhere) |(/4- g + |/~ g\) and i(/+ # — |/- #|).
Hence they are integrable.

(13.7.5) For a function he ^ (resp. g e &*)to be integrable, it is necessary and
sufficient that n*(h) < + oo (resp. p*(g) > — oo).

For if ju*(/z) < + oo, then for any e > 0 there exists u e JTR(X) such that
u £ h and n*(h - M) = ju*(/z) - /*(«) ^ e, by (13.5.1 ) and (13.5.3). The result
now follows from the definition of integrable functions.

A subset A of X is said to be inferrable if its characteristic function (pA is
integrable, or equivalently if ju*(A) and ju*(A) are finite and equal. Their

common value J <pA d/j is then denoted by ^u(A) and is called the measure

of A. We have /x(0) = 0. The negligible sets are the same as the integrable
sets with measure zero. If A is integrable, then so is every set B for which
A n (JB and B n (JA are negligible^ and ju(B) =

(13.7.6)    If A and B are integrable sets, then A u B, A n B and A n (JB are
integrable.

This follows from the formulas (12.7.3).

(13.7.7)    Every compact set is integrable. For an open set U to be integrable
it is necessary and sufficient that ju*(U) < H-oo. In particular, every relatively
compact open set is integrable.

This follows immediately from (13.7.5) and (12.7.4).

Example

(13.7.8)   With respect to Lebesgue measure on R, every bounded interval
with endpoints a, b is integrable and its measure is \b — a|, by virtue ofion /is integrable if the functions equivalent to /and defined